This video suggests that a universe exists for each different possibility in the world. However, doesn't that mean that there must be a universe in which this theory has been disproved? Wouldn't that make this way of understanding different dimensions nonsensical?
No. There would be no valid disproof. We generally hold that it is not possible to validly disprove something which is true. So if there are ten dimensions, any proof which concludes that there are not ten dimensions has a flaw somewhere in it. This is very different from some universes not being able to prove that there are ten dimensions. In some universes, I can imagine that an a-Ten-Dimensionalist could say:
There is neither sufficient evidence, nor reason, to believe that there are ten dimensions.
Such a person need not assert that there aren't ten dimensions. Should he/she do that, he/she would be in error.
Each possibility, from what range of possibilities?
By the very nature of the notion of something which holds "in a world", it does not make obvious sense that something which holds of a world can be a property of the collection of all worlds. To make a (rough) analogy: it is not clear that if one country does not (politically speaking) formally recognise the existence of any other country, that this should mean anything about the existence of any other country.
The Many Worlds Hypothesis in quantum mechanics is that any possible measurable outcome of an experiment, is in principle realised in some parallel 'world'. The existence of parallel worlds is not, however, an outcome which may be experimentally tested by any convincing means that we know of. (If we could, the Many Worlds Hypothesis would be much less controversial — it would simply be accepted as true or false, at least with as much confidence as other testable assertions.)
The [realist interpretation of the] many worlds hypothesis does not disprove itself, but its lack of verifiability and its ad hoc use in cosmology -- a way to solve a problem where it looks there would need to be a creation event should give us reasons to doubt the truth of this hypothesis.
Moreover, a question arises as to what the existence of an infinite number of universes would mean for the nature of existence in any of them.